Factorization of numbers with truncated Gauss sums at rational arguments

Abstract

Factorization of numbers with the help of Gauss sums relies on an intimate relationship between the maxima of these functions and the factors. Indeed, when we restrict ourselves to integer arguments of the Gauss sum we profit from a one-to-one relationship. As a result, the identification of factors by the maxima is unique. However, for non-integer arguments, such as rational numbers, this powerful instrument to find factors breaks down. We develop new strategies for factoring numbers using Gauss sums at rational arguments. This approach may find application in a recent suggestion to factor numbers using a light interferometer discussed in this issue (V. Tamma et al., J. Mod. Opt. in press).

Keywords:

• Gauss sum
• factorization
• rational arguments

Acknowledgments

We thank M. Jakob, M. Štefańǎk and M.S. Zubairy for many fruitful discussions on this topic. This research was partially supported by the Max Planck Prize of WPS awarded by the Humboldt Foundation and the Max Planck Society.

Notes

Notes

1. It is interesting to note that already in 1924–that is, before the advent of modern quantum mechanics introduced by Heisenberg and Schrödinger–Gregor Wentzel proposed the path integral in a paper entitled ‘Zur Quantenoptik’. Unfortunately, he never followed up on his idea.

2. One of the referees has speculated in his report that if each challenge would carry a phase we may find a new quantum mechanics of life. Indeed, Pascual Jordan 18 and Erwin Schrödinger 19 have argued that life might be intimately connected to quantum mechanics.